On existence–uniqueness results for proportional fractional differential equations and incomplete gamma functions

Abstract

In this article, we employ the lower regularized incomplete gamma functions to demonstrate the existence and uniqueness of solutions for fractional differential equations involving nonlocal fractional derivatives (GPF derivatives) generated by proportional derivatives of the form $$ D^{\rho }= (1-\rho )+ \rho D, \quad \rho \in [0,1], $$ D ρ = ( 1 − ρ ) + ρ D , ρ ∈ [ 0 , 1 ] , where D is the ordinary differential operator.

On the Initial Value Problems for Caputo-Type Generalized Proportional Vector-Order Fractional Differential Equations

“…where Φ * j := sup t∈J Φ j (t). Therefore, from inequality (17) it follows Az n − Az C r → 0 as n → ∞. Thus, the operator A is continuous.…”

“…In addition to the most popular fractional derivatives, the Liouville-Caputo, Riemann-Liouville, and Hadamard fractional operators, the emergence of new ones, such as the Hilfer, Katugampola, Caputo-Fabrizio, and generalized proportional fractional operators, has enriched the research in the topic; see [12][13][14][15][16][17] and the references therein.…”

Untitled

“…In this section we give some definitions and lemmas from the theory of fractional calculus.We start by defining Generalized Proportional fractional integrals and derivatives. these definitions are adopted from [12], [5].…”

Stability of Generalized Proportional Caputo Fractional Differential Equations by Lyapunov Functions

“…Note that the GPFE is similar to the so-called tempered fractional derivative. For some results concerning the GPFR and the differential equations with the GPFD, as well as its applications, we refer the reader to [9][10][11][12][13]. However, the study of the stability properties of the solutions of fractional differential equations with the GPFD is at its initial stage (see, for example, [14]).…”